On some conjectures on VOAs Yuji Tachikawa February 1, 2013 In [1], a lot of mathematical conjectures on VOAs were made. Here, we’ll provide a more mathematical translation, along the lines of [2]. I’m sure that I used many notations not common in the VOA literature; please point them out. I’m willing to make this as readable as possible for VOA theorists.

1

Category V

In the following, a VOA always stands for vertex operator super-algebra V = ⊕n Vn , Vn = Vn,+ ⊕ Vn,− . The grading n is over integers and half-integers. The part V± = ⊕n Vn,± are called bosonic and fermionic, respectively. Given two VOAs V1 and V2 , we denote its product by V1 × V2 . We denote the zero mode of a field φ(z) by φ0 . Given a VOA V with a homomorphism φ : gˆ → V, let us denote by DS(V, g, e) obtained by the quantum Drinfeld-Sokolov reduction of V with respect to the affine algebra gˆ and a nilpotent element e of g. Again, given a VOA V = ⊕n Vn with a homomorphism φ : gˆ → V, we define its character ch V(z) for z ∈ G where G is a simply-connected group for the Lie algebra g by X ch V(z) = q n StrVn z. (1.1) n

We consider the following category V: • Its objects are VOAs of the form n Y

gˆi ,−h∨ (gi )

(1.2)

i=1

where gi is a simple Lie algebra, gˆk is the affine Lie algebra for g regarded as a VOA with k its level, and h∨ is the dual Coxeter number. We allow the case when n = 0, which we denote by 1. In the following, we simply denote by gˆ a VOA of the form (1.2). 1

• An element in Hom(gˆ, gˆ0 ) is a triple (V, φ, φ0 ) where V is a VOA and φ, φ0 are homomorphisms φ : gˆ → V, φ0 : gˆ0 → V. (1.3) Writing φ, φ0 explicitly is tedious, and we often just denote V ∈ Hom(g, g0 ). • Given V ∈ Hom(gˆ, gˆ0 ) and V 0 ∈ Hom(gˆ0 , gˆ00 ), its composition V 0 ◦ V is defined as a ‘VOA version of holomorphic symplectic quotient’ as will be defined later. Before defining the composition, let us discuss easier properties. • V is a symmetric monoidal category. Given two objects g and g0 , its product is g × g0 . Similarly, given (V1 , φ1 , φ01 ) ∈ Hom(gˆ1 , gˆ01 ) and (V2 , φ2 , φ02 ) ∈ Hom(gˆ2 , gˆ02 ), its product is (V1 × V2 , φ1 × φ2 , φ01 × φ02 ). • V ‘has duality’ in the following technical sense. There is a canonical identification Hom(gˆ, gˆ0 × gˆ00 ) ' Hom(gˆ × gˆ00 , gˆ0 )

(1.4)

given as follows. Given an element (V, φ, φ0 × φ00 ) of the Hom on the left, we assign an element (V, φ × (φ00 ◦ π), φ0 ) on the right, where π is an automorphism of gˆ obQ tained by extending the automorphism of g = i gi which sends a finite dimensional representation to its dual. Now, let us define the composition of morphisms. For this, we define an operation Quot(V, gˆ−2h∨ (g) , φ) which gives a new VOA given a VOA V with a homomorphism φ: φ : gˆ−2h∨ (g) → V.

(1.5)

Note that the level here is −2h∨ (g). To define Quot, we first introduce a ghost VOA BC(g), generated by fermionic fields bA in BC(g)1,− and cA in BC(g)0,− for A = 1, . . . , dim g with the OPE δBA bA (z)cB (w) ∼ . (1.6) z−w A This has a subalgebra gˆ+2h∨ (g) . Denote by JVA (z) and Jghost the affine g currents of V and BC(V) respectively. We define jBRST (z) =

X A

1 A (cA JVA (z) + cA Jghost (z)). 2

(1.7)

Then d = jBRST,0 is nilpotent, thanks to the condition on the level of gˆ in V. We take the subspace W ⊂ V × BC(g) (1.8) defined by W=

\

A (Ker bA 0 ∩ Ker(Jtotal ))

A

2

(1.9)

A A where Jtotal = JVA + Jghost . We can check that the differential d acts within W, and finally we define Quot(V, gˆ−2h∨ (g) , φ) = H(W, d). (1.10) Q Quot for products i gˆi,−2h∨ (gi ) is defined in a completely similar way. This is like a VOA version of (holomorphic) symplectic quotient: (1.9) is like setting the moment map to be zero, and (1.10) is like taking the quotient. Then, given (V1 , φ01 , φ1 ) ∈ Hom(gˆ0 , gˆ) and (V2 , φ2 , φ002 ) ∈ Hom(gˆ, gˆ00 ), its composition is given by

(V2 , φ2 , φ002 ) ◦ (V1 , φ01 , φ1 ) = (Quot(V1 × V2 , gˆ−2h∨ (g) , φ1 + (φ2 ◦ π)), φ01 , φ002 ).

(1.11)

Note that this is possible because φ1 + (φ2 ◦ π) defines a homomorphism from gˆ−2h∨ (g) to V1 × V2 . Now, we have cheated at one point, which needs to be stated as a conjecture here: Conjecture 1. There is a VOA Idˆg ∈ Hom(gˆ, gˆ) acting as an identity under the composition defined above. Most probably, in the decomposition Idˆg = ⊕∞ n=0 V0 , V0 is isomorphic to C[G] where G is the algebraic group over C of type g. In particular, V0 is infinite dimensnional.

2

Functor ηg

We use another category B, i.e. the objects are closed oriented one-dimensional manifolds (i.e. disjoint unions of multiple S 1 s) and a morphism from B1 to B2 is a two-dimensional oriented manifold C whose boundary is B1 t (−B2 ). B is a symmetric monoidal category with duality under the standard operations. For example, we have U=

∈ Hom(S 1 , ∅),

(2.1)

V =

∈ Hom(S 1 t S 1 , ∅),

(2.2)

W =

∈ Hom(S 1 t S 1 t S 1 , ∅).

(2.3)

Conjecture 2. For each simple, simply-laced Lie algebra g. Then there is a functor ηg : B → V

(2.4)

with ηg (S 1 ) = gˆ−h∨ (g) , such that the VOAs Ug = ηg (

),

Vg = ηg (

),

satisfy the following properties: 3

Wg = ηg (

)

(2.5)

1. Vg is the identity Idg . 2. Ug is obtained from Vg via the Drinfeld-Sokolov reduction, ) = DS(ηg (

ηg (

), gˆ, eprin )

(2.6)

where eprin is a principal nilpotent element of g. 3. Wg has central charge c = (3 − 2h∨ (g)) dim g − rank g

(2.7)

and the following character: ch Wg (z1 , z2 , z3 ) =

K0 (z1 )K0 (z2 )K0 (z3 ) X χλ (z1 )χλ (z2 )χλ (z3 ) . Kprin χλ (q ρ ) λ

(2.8)

Here, (z1 , z2 , z3 ) ∈ G3 which is the exponential of the action of g3 on Wn , the sum is over all the irreducible representation λ of g, χλ is the character, K0 is the character of z in the representation m ∞ n K0 = ⊕∞ m=0 Sym (⊕n=1 (q g)),

and Kprin =

∞ rank Y Yg n=1 i=1

(2.9)

1 1 − q di +n

(2.10)

where di is the i-th exponent of g plus 1, and the element q ρ ∈ G is defined as usual by the Weyl vector. For example, q ρ = diag(q 1/2 , q −1/2 ) when g = A1 . Remark. The generating relations of B is not very complicated. Therefore, to prove that ηg exists, one just has to do the following. • Find a VOA Wg satisfying (2.7) and (2.8). • Show

ηg (

) = ηg (

),

(2.11)

where the VOA on the both sides are understood to be defined from two copies of Wg by (1.11). • Show ηg (

) := DS(ηg (

is the identity of Hom(gˆ, gˆ). 4

), g, eprin )

(2.12)

Another remark. From the definition of the operation Quot, it is not too difficult to show that given φ : gˆ−h∨ (g) → V and φ0 : gˆ−h∨ (g) → V 0 , the character of W = Quot(V × V 0 , gˆ−2h∨ (g) , φ + φ0 ◦ π) is given by Z ch W = ch V(z) ch V 0 (z −1 )K0 (z)−2 [dz] (2.13) Gcpt

where the integral is taken over the compact simply-connected Lie group Gcpt of type g, and )

[dz] is the Haar measure. From this, it is easy to compute the character of ηg ( using orthogonality of the irreducible characters, assuming (2.8). More generally, denote by Cn a sphere with n S 1 boundaries. Then we easily see that X Qn K0 (zi )χλ (zi ) i=1 . (2.14) ch ηg (Cn )(z1 , . . . , zn ) = ρ n−2 (K prin χλ (q )) λ Note also that, from (2.6), we have ηg (Cn−1 ) = DS(ηg (Cn ), g, eprin ).

(2.15)

This plays nicely with the formula (2.14) due to the following general fact: Given φ : gˆ → V, the character of W = DS(V, eprin ) is obtained by ch V(z) . ) ch W = Kprin ( K0 (z) z→qρ

(2.16)

Slightly generalized version of ηg . We slightly generalize the definition of ηg , by allowing surfaces with points labeled by a nilpotent element e of g on the 2d surfaces. More precisely, we consider a category Bg where the objects are still one-dimensional manifolds, and the morphisms are 2d surfaces with points each labeled by a nilpotent element of g. We do not allow points on the boundary of the 2d surface. We then extend the functor ηg : B → V to a functor Bg → V by defining e.g. e

ηg (

) = DS(ηg (

), gˆ, e).

(2.17)

In general, to define ηg for a punctured surface, we replace each marked point by a boundary. We then perform, for each gˆ associated to the newly introduced boundary, the quantum Drinfeld-Sokolov reduction by the nilpotent e. We immediately see that a point labeled by e = 0 is equivalent to a boundary. Also, due to (2.6), having a point labeled by eprin is equivalent to having no point at all. 5

3

Examples

Not much is known about Wg . To state what is known, let us introduce a few VOAs. First, given a symplectic vector space V over C, denote by SB(V ) a VOA such that SB(V ) = ⊕∞ n=0 Vn , and

V0 = C1,

V1/2,+ ' V,

(3.1)

hv, wi (3.2) z where h·, ·i is the symplectic pairing. This has the central charge v(z)w(0) '

for v, w ∈ V ' V1/2 c = −(dimC V )/2. Let us also define L(gˆ, k) to be the VOA based on the irreducible, vacuum representation of the affine Lie algebra gˆ with level k. Conjecture 3.

• For g = A1 , we have ηg (

) = SB(V1 ⊗ V2 ⊗ V3 ),

(3.3)

) = L(eb6 , −3).

(3.4)

where Vi ' C2 . • For g = A2 , we have ηg ( • For g = An−1 , we have ηg (

[n-1,1]

) = SB(V ⊗ W ∗ ⊕ W ⊗ V ∗ ).

(3.5)

Here, [n − 1, 1] is a nilpotent element consisting of two Jordan blocks of size n − 1, 1 (i.e. an element on the subregular orbit) and V, W ' Cn . Combined with (3.4), we have d , e = [2, 1]) = SB(V ⊗ W ∗ ⊕ W ⊗ V ∗ ), DS(L(eb6 , −3), sl(3) (3.6) −3 i.e. a variant of the quantum Drinfeld-Sokolov reduction of the affine E6 algebra gives a free boson. Note that to complete the proof of the existence of the functor ηg for g = A1 and g = A2 , we just have to show (2.11) and (2.12), starting from Wg given by (3.3) and (3.4). In particular, as for the relation (2.11), we have the following conjecture.

6

Conjecture 4. For g = A1 ,

[ −2). ) = L(so(8),

ηg (

(3.7)

Note that sl(2)4 ⊂ so(8), and the outer automorphism of so(8) permutes four copies of sl(2). Combined with (3.3), we have a free-field realization d , φ) [ −2) = Quot(SB(V1 ⊗ V2 ⊗ V3 ) × SB(V3 ⊗ V4 ⊗ V5 ), sl(2) L(so(8), −4

(3.8)

where φ : sl(2) → SB(V1 ⊗ V2 ⊗ V3 ) × SB(V3 ⊗ V4 ⊗ V5 ) comes from the action of SL(2) on V3 . Finally, we have the following, as a further generalization of (3.4) and (3.7) : Conjecture 5.

• For g = A1 , [1,1]

[1,1]

ηA1 ( [1,1]

[ −2). ) = L(so(8),

(3.9)

) = L(eb6 , −3).

(3.10)

[1,1]

Note that this is equivalent to (3.7). • For g = A2 , [1,1,1]

ηA2 ( [1,1,1]

[1,1,1]

Note that this is equivalent to (3.4). • For g = A3 , [2,2]

) = L(eb7 , −4).

ηA3 ( [1,1,1,1]

(3.11)

[1,1,1,1]

• For g = A5 , [3,3]

) = L(eb8 , −6).

ηA5 ( [2,2,2]

(3.12)

[1,1,1,1,1,1]

Note that the number of points on S 2 , and the Jordan block structures of the nilpotent elements labeling the points, follow the structure of the extended Dynkin diagrams of D4 , E6,7,8 , respectively. 7

References [1] C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees, “Infinite Chiral Symmetry in Four Dimensions,” arXiv:1312.5344 [hep-th]. [2] G. W. Moore and Y. Tachikawa, “On 2D TQFTs Whose Values are Holomorphic Symplectic Varieties,” arXiv:1106.5698 [hep-th].

8